识别Rayleigh-Stokes方程源项的分数阶Landweber迭代正则化方法

Abstract

Abstract:

In this paper, the inverse problem of identifying the unknown sources for the Rayleigh-Stokes equation with a Riemann-Liouville fractional derivative in time is considered. We prove that such a problem is ill-posed and apply the fractional Landweber method to solve this inverse problem. Based on the results of conditional stability, under the priori and posteriori regularization parameters choice rules, the error estimates between the exact solution and the regularization solution are given respectively. Finally, several numerical examples are given to illustrate the effectiveness and feasibility of these methods.

Key words: Rayleigh-Stokes equation, Inverse problem, Identifying unknown source problem, Fractional Landweber iterative regularization method

CLC Number:

O175

Fan Yang,Qianchao Wang,Xiaoxiao Li. Fractional Landweber Iterative Regularization Method to Identify Source Term for the Rayleigh-Stokes Equation[J].Acta mathematica scientia,Series A, 2021, 41(2): 427-450.

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Figures/Tables 21

References 46

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α				α = 0.2	α = 0.5	α = 0.9
η(f)	ε=0.01	Landweber	BD	0.0099	0.0083	0.0075
			C-N	0.0072	0.0052	0.0041
		分数阶Landweber	BD	0.0088	0.0076	0.0061
			C-N	0.0063	0.0048	0.0039

	ε=0.005	Landweber	BD	0.0042	0.0039	0.0034
			C-N	0.0041	0.0035	0.0025
		分数阶Landweber	BD	0.0038	0.0035	0.0021
			C-N	0.0031	0.0028	0.0011

	ε= 0.001	Landweber	BD	0.0025	0.0019	7.9678e-04
			C-N	0.0020	0.0011	5.5690e-04
		分数阶Landweber	BD	0.0018	0.0010	7.1943e-04
			C-N	0.0011	8.7631e-04	4.6982e-04

α				α = 0.2	α = 0.5	α = 0.9
迭代步数(m)	ε = 0.01	Landweber	BD	24027	9658	30
			C-N	25556	31634	37652
		分数阶Landweber	BD	19683	6514	21
			C-N	21423	23963	29685

	ε = 0.005	Landweber	BD	46820	21349	34
			C-N	27941	30652	39541
		分数阶Landweber	BD	39870	12981	29
			C-N	19685	23916	33921

	ε = 0.001	Landweber	BD	79686	38679	44
			C-N	35719	42387	52802
		分数阶Landweber	BD	68765	27695	38
			C-N	26985	34796	44348

α				α = 0.2	α = 0.5	α = 0.9
CPU时间(unit: s)	ε = 0.01	Landweber	BD	480.54	193.16	0.60
			C-N	511.12	632.68	753.04
		分数阶Landweber	BD	393.66	130.28	0.42
			C-N	428.46	479.26	593.70

	ε = 0.005	Landweber	BD	936.40	426.98	0.68
			C-N	558.82	613.04	790.82
		分数阶Landweber	BD	797.40	259.62	0.58
			C-N	393.70	478.32	678.42

	ε = 0.001	Landweber	BD	1593.72	773.58	0.88
			C-N	714.38	847.74	1056.04
		分数阶Landweber	BD	1375.30	553.90	0.76
			C-N	539.70	695.92	886.96

α				α = 0.2	α = 0.5	α = 0.9
η(f)	ε = 0.01	Landweber	BD	0.0243	0.0128	0.0088
			C-N	0.1039	0.0864	0.0747
		分数阶Landweber	BD	0.0208	0.0106	0.0075
			C-N	0.0931	0.0705	0.0628

	ε = 0.005	Landweber	BD	0.0155	0.0116	0.0076
			C-N	0.0838	0.0725	0.0634
		分数阶Landweber	BD	0.0191	0.0103	0.0063
			C-N	0.0786	0.0596	0.0413

	ε = 0.001	Landweber	BD	0.0033	0.0019	8.6476e-04
			C-N	0.0589	0.0362	0.0295
		分数阶Landweber	BD	0.0021	0.0011	6.5483e-04
			C-N	0.0412	0.0268	0.0105

α				α = 0.2	α = 0.5	α = 0.9
迭代步数(m)	ε = 0.01	Landweber	BD	22348	6987	29
			C-N	3717	5816	7425
		分数阶Landweber	BD	16534	3768	19
			C-N	2674	4168	6123

	ε = 0.005	Landweber	BD	34258	12368	33
			C-N	6670	8126	11198
		分数阶Landweber	BD	22369	6879	26
			C-N	4396	6021	8934

	ε = 0.001	Landweber	BD	79463	39654	41
			C-N	15285	86956	131965
		分数阶Landweber	BD	54986	21685	33
			C-N	13210	69663	113824

Fractional Landweber Iterative Regularization Method to Identify Source Term for the Rayleigh-Stokes Equation

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α				α = 0.2	α = 0.5	α = 0.9
CPU时间(unit: s)	ε = 0.01	Landweber	BD	446.96	139.74	0.58
			C-N	74.34	116.32	148.50
		分数阶Landweber	BD	330.68	75.36	0.38
			C-N	53.48	83.36	122.46

	ε = 0.005	Landweber	BD	685.16	247.36	0.66
			C-N	133.4	162.52	223.96
		分数阶Landweber	BD	447.38	137.58	0.52
			C-N	87.92	120.42	178.68

	ε = 0.001	Landweber	BD	1589.26	793.08	0.82
			C-N	305.70	1739.12	2639.3
		分数阶Landweber	BD	1099.72	433.70	0.66
			C-N	264.20	1393.26	2276.48

α				α = 0.2	α = 0.5	α = 0.9
η(f)	ε = 0.01	Landweber	BD	0.0287	0.0196	0.0100
			C-N	0.1867	0.2375	0.2735
		分数阶Landweber	BD	0.0234	0.0165	0.0008
			C-N	0.1529	0.2143	0.2568

	ε = 0.005	Landweber	BD	0.0176	0.0102	0.0050
			C-N	0.1679	0.2139	0.2228
		分数阶Landweber	BD	0.0153	0.0088	0.0016
			C-N	0.1428	0.1796	0.1928

	ε = 0.001	Landweber	BD	0.0068	0.0036	0.0010
			C-N	0.0924	0.1428	0.1522
		分数阶Landweber	BD	0.0053	0.0018	2.5638e-04
			C-N	0.0723	0.1256	0.1347

α				α = 0.2	α = 0.5	α = 0.9
迭代步数(m)	ε = 0.01	Landweber	BD	47906	21398	29
			C-N	4837	8878	19844
		分数阶Landweber	BD	31695	10321	18
			C-N	2968	5367	16482

	ε = 0.005	Landweber	BD	74840	46709	33
			C-N	10244	168704	52632
		分数阶Landweber	BD	62390	30987	28
			C-N	7688	11326	40986

	ε = 0.001	Landweber	BD	121803	76895	42
			C-N	238323	638825	788528
		分数阶Landweber	BD	98465	54326	37
			C-N	196524	419168	568238

α				α = 0.2	α = 0.5	α = 0.9
CPU时间(unit: s)	ε = 0.01	Landweber	BD	958.12	427.96	0.58
			C-N	96.74	177.56	396.88
		分数阶Landweber	BD	639.30	206.42	0.36
			C-N	59.36	107.34	329.64

	ε = 0.005	Landweber	BD	1496.80	934.18	0.66
			C-N	204.88	3374.08	1052.64
		分数阶Landweber	BD	1247.80	619.74	0.56
			C-N	153.76	226.52	819.72

	ε = 0.001	Landweber	BD	2436.06	1537.90	0.84
			C-N	4766.46	12776.50	15770.56
		分数阶Landweber	BD	1969.30	1086.52	0.74
			C-N	3930.48	8383.36	11364.76